{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# p07: Accuracy of periodic spectral differentiation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format='svg'\n",
    "from numpy import zeros,pi,inf,linspace,arange,tan,sin,cos,exp,abs,dot\n",
    "from scipy.linalg import toeplitz,norm\n",
    "from matplotlib.pyplot import figure,subplot,semilogy,title,xlabel,ylabel,axis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
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   "source": [
    "# Set up grid and differentiation matrix:\n",
    "Nmax = 50\n",
    "E = zeros((4,Nmax/2-2))\n",
    "for N in range(6,Nmax+1,2):\n",
    "    h = 2.0*pi/N; x = h*linspace(1,N,N);\n",
    "    col = zeros(N)\n",
    "    col[1:] = 0.5*(-1.0)**arange(1,N)/tan(arange(1,N)*h/2.0)\n",
    "    row = zeros(N)\n",
    "    row[0] = col[0]\n",
    "    row[1:] = col[N-1:0:-1]\n",
    "    D = toeplitz(col,row)\n",
    "    \n",
    "    v = abs(sin(x))**3\n",
    "    vprime = 3.0*sin(x)*cos(x)*abs(sin(x))\n",
    "    E[0][N/2-3] = norm(dot(D,v)-vprime,inf)\n",
    "    \n",
    "    v = exp(-sin(x/2)**(-2))   # C-infinity\n",
    "    vprime = 0.5*v*sin(x)/sin(x/2)**4\n",
    "    E[1][N/2-3] = norm(dot(D,v)-vprime,inf)\n",
    "    \n",
    "    v = 1.0/(1.0+sin(x/2)**2)     # analytic in a strip\n",
    "    vprime = -sin(x/2)*cos(x/2)*v**2\n",
    "    E[2][N/2-3] = norm(dot(D,v)-vprime,inf)\n",
    "    \n",
    "    v = sin(10*x)\n",
    "    vprime = 10*cos(10*x)   # band-limited\n",
    "    E[3][N/2-3] = norm(dot(D,v)-vprime,inf)\n",
    "\n",
    "\n",
    "titles = [\"$|\\sin(x)|^3$\", \"$\\exp(-\\sin^{-2}(x/2))$\", \\\n",
    "          \"$1/(1+\\sin^2(x/2))$\", \"$\\sin(10x)$\"]\n",
    "figure(figsize=(9,10))\n",
    "for iplot in range(4):\n",
    "    subplot(2,2,iplot+1)\n",
    "    semilogy(arange(6,Nmax+1,2),E[iplot][:],'o-')\n",
    "    title(titles[iplot])\n",
    "    xlabel('N')\n",
    "    ylabel('error')\n",
    "    axis([0,Nmax,1.0e-16,1.0e3])"
   ]
  }
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